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Full text of "Analytical Mechanics"

CENTER OF MASS AND MOMENT OF INERTIA     1<
2. Find the center of mass of a right circular cone whose densi varies inversely as the square of the distance from the apex, the distan being measured along the axis.
dm = r  iry2  dx
= ^'7T
Tlira
dx
where n is the density at a unit distance from the apex. Therefore
1    ~h
 (   xdx
n
h'2 h '' 2
 f dx
S n
FIG. 82.
PROBLEMS.
1.   Find the center of mass of a right circular cone, the density of whi varies inversely as the distance from the vertex.
2.   Find the center of mass of a circular plate, the density of whi varies as the distance from a point on the circumference.
3.   Find the center of mass of a cylinder, the density of which var with the nth power of the distance from one base.
4.   Find the center of mass of a quadrant of an ellipsoid.
6.  Find the center of mass of a hemisphere, the density of which var as the distance from the center.
119.   Center of Mass of a Number of Bodies.  Let mi; w
etc., be the masses and Xi, x2, etc., be the ^-coordinates of ti centers of mass of the individual bodies. Then if x denot the re-coordinate of the center of mass of the entire syste we can write
xdm